Normalized saddle solutions for a mass supercritical Choquard equation

We investigate the existence of saddle type normalized solutions for the nonlinear Choquard equation:{−Δu−λu=(Iα⁎F(u))F′(u), in RN∫RN|u|2dx=a2,u∈H1(RN). Here N≥1, a>0 is given in advance, Iα is the Riesz potential of order α∈(0,N) and the unknown parameter λ appears as a Lagrange multiplier. In a mass supercritical setting on F, we prove the existence of a couple (uaG,λaG)∈H1(RN)×R− of saddle solutions for any a>0 and for given finite Coxeter group G with its rank k≤N. Our method is to combine the concentration compactness principle with a minimax procedure in the saddle type symmetric subspace, which gives a variational framework of constructing normalized saddle solutions for the Choquard equation.